
The first equality is Corollary 5.2.5 in [1]}. The second equality is the usual chain rule for KL-divergences. The third equality recognizes that conditioned on \(\tau <t\) , \(\tilde{Z}_t =0\)  almost surely under both \(\mathbb {P}(\cdot )\)  and \(\mathbb {Q}(\cdot )\)  and so \(D_{\rm KL}\left( \mathbb {P}(\tilde{Z}_t \in \cdot \mid \tilde{Z}_{1:(t-1)} ) \, || \, \mathbb {Q}(\tilde{Z}_t \in \cdot \mid \tilde{Z}_{1:(t-1)} ) \right) =0\) . The fourth equality uses that KL-divergences are non-negative and applies the monotone convergence theorem to interchange the limit and expectation.
